OpenYear of origin: 1992

Posted online: 2020-02-18 17:23:53Z by Henrik Shahgholian45

Cite as: P-200218.1

Isaac Newton in his *Principia* (first book Ch. 12, Theorem XXXI)
asserts that spherical shells exert no gravitational force in the cavity of the shell.
This result was later extended to (ellipsoidal) homoeoid by P.-S. Laplace, using computation, and soon after by J. Ivory, using a more geometric approach; a homoeoid is
the domain bounded by two homothetic ellipsoids with a common center.

Ivory's proof is based on the following simple observation (for ellipsoids). Let $E = \{x: \ \sum_j a_j x_j^2 < 1 \}$ ($a_j > 0$) be an ellipsoids in ${\mathbb R}^n$ ($n\geq 2)$ (Ellipse in dimension 2), centred at the origin. Set $$ E_t := tE, \qquad t > 1 \qquad (\hbox{dilation}).$$ Let $x \in E$, and $L$ be any line through $x$. Then

i) $L$ cuts $\partial E$, at 2 points: $p, q \in \partial E$, on each side of $x$,

ii) $L$ cuts $\partial E_t$, at 2 points: $p_t, q_t \in \partial E_t$, on each side of $x$.

** Ivory's Lemma:**
$$|p-p_t| = |q - q_t|.$$

Using spherical coordinates with center $x$ we have $$ E_t \setminus E = \{y: \ y = x + r\omega , \ \rho (\omega) < r < \rho_t (\omega) \} , $$ where $\rho, \rho_t$ are respectively the equation of the ellipsoids $E, E_t$ in spherical coordinates with pole at $x$.

Define the Newtonian potential of $M \subset {\mathbb R}^n$, with density $c_n$, by $$ V_M (x) := c_n\int_{M} \frac{ dy}{|y-x|^{n-2}} . $$ In dimension 2 one needs to take logarithmic kernel. Here we have assumed $M$ is such that the integral exists, and we have chosen $c_n$ as a normalization factor such that $$ \Delta V_M := \nabla \cdot \nabla V_M= -\chi_M. $$ $ \nabla V_M$ is the gravitational field of $M$. Here $\Delta $ is the Laplace operator, and $\chi_M $ the characteristic function of $M$. For $x \in E $ we have

$$ \frac{1}{(n-2)c_n}\nabla V_{E_t \setminus E} (x) : = \int_{E_t \setminus E} \frac{(y-x) dy}{|y-x|^n} = \int_{{\mathbb{S}}^{n-1}} \int_{\rho(\omega)}^{ \rho_t (\omega)} \frac{ r\omega r^{n-1} dr d\omega}{ r^n} $$

$$
= \int_{{\mathbb{S}}^{n-1}} \omega R(\omega) d\omega ,
$$
where
$$R(\omega ) = \left[ \rho_t (\omega) - \rho (\omega) \right] , \qquad y - x = r\omega,$$
and $d\omega$ is the differential solid angle; in ${\mathbb R}^3$, $d\omega = \sin \theta d \theta$, $0\leq \theta \leq \pi$.
By Ivory's Lemma $R(\omega) = R(-\omega)$, and therefore
$$\int_{{\mathbb{S}}^{n-1}} \omega R(\omega) d\omega = 0.$$
Hence the gravitational field $\nabla V_{E_t \setminus E} (x)$ is zero in the cavity, i.e.
$$
0= \nabla V_{E_t \setminus E} (x) ,
$$
for $x \in E$. **Hence no-gravity in the cavity.**

Rewriting the above we have $$ 0= \nabla V_{E_t \setminus E} (x) = \nabla V_E (x) - \nabla V_{E_t} (x) $$ and hence $$ \nabla V_E (x) = \nabla V_{E_t} (x) = t \nabla V_E (x/t) , \qquad \forall \ x \in E. $$ Therefore $\nabla V_E$ is degree one homogeneous in $E$. By continuity, each component must be linear. Hence $V_E$ is a quadratic polynomial $Q_E$ inside $E$.

By rigid motion of $E$, we can rewrite $Q_E$ as $$ Q_E= K_E - P_2 , \qquad \hbox{where } \quad P_2 = \sum_i b_{i} x_i^2$$ $$K_E >0 \ \hbox{ is a constant, } \qquad b_i > 0, \qquad 2 \sum_i b_i = 1.$$ We have used $ K_E = V_E(0) > 0$, and that gravitational force decreases as we move inside from the boundary, hence $$ \partial_{ii} V_{E} = \partial_{ii} Q_{E} = 2b_{i} > 0.$$

A result by Dive ( [4] 1931), and Di-Benedetto-Friedman ([5] 1986) states that

*Ellipsoids are the only bounded domains with Newtonian potentials being quadratic polynomials inside the domain.*

**Proof **(sketch):
Let $D$ be the domain whose Newtonian potential $V_D$ is a quadratic polynomial
$$V_D= K_D - P_2 , \qquad \hbox{inside }D .$$

There exists a smallest ellipsoid $E \supset D$ such hat $V_E$ is the polynomial $$V_E= K_E - P_2 \qquad \hbox{inside } E . $$ Dive did this by "simple" algebra using ellipsoidal coordinates. Di-Benedetto-Friedman used an iterative tracking method.

Obviously $K_E > K_D$, since $E \supset D$. For $z \in \partial E \cap \partial D$, a touching point of the boundaries, we have $$ V_{E\setminus D} = ( K_E - P_2 ) - (K_D - P_2 ) = K_E - K_D = \ Constant \quad \hbox{inside } D , $$ and in particular $$ \nabla V_{E\setminus D} = 0 \qquad \hbox{inside } D. $$

Compute the gravitation force at $z$ in the direction ${\bf a } -z$, and recall that $$ \nabla V_{E\setminus D} = 0 \qquad \hbox{inside } D. $$ Hence $$ 0= ({\bf a } -z) \cdot \nabla V_{E\setminus D} (z) = (n-2)c_n\int_{E\setminus D} \frac{ ({\bf a} -z) \cdot (y - z)}{|y-z|^n} dx > 0, $$ a contradiction. Therefore $D=E$, and $D$ is an ellipsoid.

**Unbounded domains:**

*
Can we define the concept for unbounded domains, e.g. limits of ellipsoids (paraboloids, half-planes, cylindrical domains)?
*

**PDE formulation:**
Since for any ellipsoid $E$ we have $V_E = Q_E$
in $E$, we define
$$
u(x):= V_E (x) - Q_E (x) \qquad x\in {\mathbb R}^n,
$$
and obtain a global solution to the obstacle problem.
$$
\Delta u = \chi_{ E^c}, \qquad u=0 \quad \hbox{in } E ,
$$
$$|u (x) | \approx C|x|^2, \qquad \hbox{for $ |x| $ being large} .$$

This definition generalizes the above concept to limits of a sequences of ellipsoids:

A solution to the so-called obstacle problem $$ \Delta u = \chi_{ \Omega }, \qquad u=0 \quad \hbox{in } \Omega^c , $$ $$|u (x) | \approx C|x|^2, \qquad \hbox{for $ |x| $ \ large} .$$ is called a global solution. The set $\Omega^c= \{ u = 0\}$ is called the coincidence set.

It is well known [3] that $u\geq 0$, so that $\Omega = \{u > 0\}$.

**Conjecture: Sh. 1992:**
The only global solutions to the obstacle problem are those whose coincidence sets are limit domains of any sequences of ellipsoids.

**Theorem:** (Eberle, Sh., Weiss, [6] preprint 2020)

Let $u $ be a global solution to the obstacle problem in ${\mathbb R}^{n}$ $(n \geq 6)$ $$ \Delta u = \chi_{\{ u > 0\}}, \qquad u \geq 0. $$ Suppose $D:= \{ u = 0\}$ is non-cylindrical in any direction. Then $ \{ u = 0\}$ is one of the following:

** i) An ellipsoid. **

** ii) A paraboloid.** See [9]

If $ \{ u = 0\}$ is cylindrical in $k$-directions, then we can reduce the problem to ${\mathbb R}^{n-k}$, where the same conclusion holds if $n-k \geq 6$.

**Ongoing work:** This problem can be solved in dimension $n=4, 5$ by adapting the technique from [6] (preprint 2020). With some further techniques, we seem to be able to treat the case of n= 3 .
For $n=2$ there is a complex function theory approach by Sakai [2], and later by Shapiro [7]. It would be interesting to see real analysis technique for this case as well.

**Open problems:** What about polynomial densities? It is known that the Newtonian potential for an ellipsoid (and hence paraboloids, whenever integrability is available) is also a polynomial inside the body. Now, if we take the most simple form of a polynomial density, say $p(x) = x_1$, then we an form the solution to the free boundary problem as above
$$
\Delta u = x_1 \chi_{\{u \neq 0 \}} \qquad \hbox{in }\mathbb R^n, \qquad |u (x)| \leq (C(1 + |x|^3).
$$
Here the set $\{u = 0 \}$ is exactly the ellipsoid.

**A question that arise is:** Is the converse true? If not is there a counterexample?
See [8] Page 1253.

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